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Reduced and Irreducible Simple Algebraic Extensions of Commutative Rings

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Reduced and Irreducible Simple Algebraic Extensions of Commutative Rings Mathematica Moravica Vol. 21, No. 1 (2017), 69–86 Reduced and irreducible simple algebraic extensions of commutative rings S.V. Mihovski Abstract. Let A be a commutative ring with identity and α be an algebraic element over A. We give necessary and sufficient conditions under which the simple algebraic extension A[α] is without nilpotent or without idempotent elements. 1. Introduction Let A be a commutative ring with identity element 1. We shall say that K is a commutative ring extension of A, or A is a subring of K, if A and K are commutative rings with common identity element and A ⊆ K. Suppose that K is a commutative ring extension of A and let α 2 K be an algebraic element overA. If α is a root of a nonzero polynomial f(x) 2 A[x] of a minimal degree n, then we shall say that f(x) is a minimal polynomial of α over the ring A. The intersection of all subrings of K, containing A and α, we shall denoted by A[α]. The ring A[α] is called a simple algebraic extension of A, which is obtained by adjoining α to A. Let f(x) be any nonzero polynomial over the ring A. If the leading coef- ficient of f(x) is a0 = 1, then f(x) is said to be a monic polynomial over A. And what is more, if the leading coefficient a0 of f(x) is a regular element in A, i.e. a0 is not zero divisor in A, then we shall say that f(x) is a regular polynomial over the ring A. Recall that the ring A is called a reduced ring if A has no nonzero nilpo- tent elements. The ring A is said to be irreducible if A has no nontrivial idempotents. A main result in [11] asserts that if A is a reduced commutative ring, f(x) is a monic minimal polynomial of α over A and the discriminant ∆(f) is a regular element in A, then the simple algebraic extension A[α] is a reduced ring. Also in [11] is proved that if A is an irreducible ring and the minimal polynomial f(x) of α is monic and irreducible over A, then the simple algebraic extension A[α] again is irreducible. So here arises the 2010 Mathematics Subject Classification. 13B25, 13B02. Key words and phrases. Commutative rings, Polynomials, Discriminants, Resultants, Simple algebraic extensions, Reduced rings, Irreducible rings. c 2017 Mathematica Moravica 69 70 Reduced and irreducible simple algebraic extensions. problem to find necessary and sufficient conditions under which the ring A[α] is reduced or irreducible. In this paper we solve these two problems in the parts 3 and 4, respectively. 2. Preliminary lemmas and definitions It is well known that every polynomial f(x) of degree n with coefficients from a field F has at most n roots in every field extension of F . Moreover, there exists a field extension F ⊇ F such that F contains exactly n roots of f(x). But this fact does not hold for the ring extensions of F . For example, let G be a direct product of m ≥ 2 cyclic groups of order n and let K = FG be the group ring of the group G over the field F . Then K is a ring extension of F and every element of G is a root of the polynomial f(x) = xn −1 2 F [x]. Thus f(x) has at least nm roots in K. Let n n−1 (1) f (x) = a0x + a1x + ··· + an−1x + an (n ≥ 1) be a polynomial over A of degree n. We shall say that the elements α1; α2; : : : ; αn form a canonical system roots of the polynomial f(x) 2 A[x] if there exists a commutative ring extension K ⊇ A such that α1; α2; : : : ; αn 2 K and (2) f(x) = a0(x − α1)(x − α2) ··· (x − αn): Every regular polynomial over A has at last one canonical system roots [9]. But example shows that over some rings A there exist polynomials that have not roots. Moreover, there exist polynomials which have roots, but they have not canonical systems of roots. For more details see [9]. Later on we shall use the following two definitions. A discriminant of the polynomial (1) we shall call the following determinant of order 2n − 1 1 a1 ::: an 0 0 ::: 0 0 a0 a1 ::: an 0 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: 0 a a ::: a ∆(f) = " 0 1 n ; n (n−1)a1 (n−2)a2 ··· an−1 0 ::: 0 0 na0 (n−1)a1 ::: 2an−2 an−1 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: 0 na0 (n−1)a1 ::: an−1 n(n−1) where " = (−1) 2 . So, for n = 1 and n = 2 we have ∆(f) = 1 and 2 ∆(f) = a1 − 4a0a2, respectively. Let m m−1 (3) g(x) = b0x + b1x + ··· + bm−1x + bm (m ≥ 1) be another polynomial over A. Then the determinant of order n + m S.V. Mihovski 71 a0 a1 ::: an 0 ::: 0 0 0 a0 a1 ::: an 0 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: R(f; g) = 0 0 ::: a0 a1 ::: an−1 an b0 b1 ::: bm 0 ::: 0 0 0 b0 b1 ::: bm 0 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: b0 b1 ::: bm−1 bm is said to be a resultant of the polynomials f(x) and g(x) (see [3, 8, 13]). If e1; e2; : : : ; ek is a full orthogonal system idempotents of the ring A, that is the ring A is a direct sum of the ideals eiA (i = 1; : : : ; k), then for the polynomials f(x); g(x) 2 A[x] we put fi(x) = eif(x) and gi(x) = eig(x). So from the definition of R(f; g) we conclude that R(f; g) = R(f1; g1) + R(f2; g2) + ··· + R(fk; gk): Likewise, ∆(f) = ∆(f1) + ∆(f2) + ··· + ∆(fk): Later on we shall use these facts without special stipulations. Let α1; α2; : : : ; αn be a canonical system roots of the polynomial (1). In [9] it is proved that m (4) R(f; g) = a0 g(α1)g(α2) ··· g(αn) for every polynomial g(x) 2 A[x], even when g(x) does not have roots. Moreover, if n ≥ 2 and f 0(x) is the prime derivative of f(x), then [10] n(n−1) n−2 0 0 0 (5) ∆(f) = (−1) 2 · a0 · f (α1) · f (α2) ··· f (αn): When A is a field, then (4) and (5) show the well known facts that R(f; g) = 0 if and only if f(x) and g(x) have common roots and f(x) has multiple roots if and only if ∆(f) = 0. For arbitrary polynomials f(x); g(x) 2 A [x] we have the following. Lemma 1. (see [9], also [8] p.159 and [13] p. 130) Let A be any commutative ring with identity. If (1) and (3) are polynomials over A, then (i) There exist polynomials '(x); (x) 2 A[x] such that deg '(x) ≤ m − 1, deg (x) ≤ n − 1 and R(f; g) = '(x)f(x) + (x)g(x): (ii) If n ≥ 2, then there exist polynomials u(x); v(x) 2 A [x] such that deg u(x) ≤ n − 2, deg v(x) ≤ n − 1 and ∆(f) = u(x)f(x) + v(x)f 0(x): So we obtain Corollary 1. [9] Let f(x) and g(x) be polynomials over the ring A. (i) If f(x) and g(x) have common roots, then R(f; g) = 0. (ii) If f(x) has multiple roots, then ∆(f) = 0. 72 Reduced and irreducible simple algebraic extensions. Examples show that the converse statements of the preceding corollary in the general case are erroneous [9]. Let S be a multiplicatively closed set of regular elements in A, that is S contains the product of every his two elements and every element of S is not zero divisor in A. Then there exists a ring of quotients S−1A with respect to S ([12], p. 146). Every element of S−1A is of the form s−1a, where s 2 S, a 2 A. Thus all elements of S are invertible in S−1A. If S is the set of all regular elements in A, then S−1A is said to be the classical ring of quotients, that we shall denote by Q(A). Lemma 2. Let A be a ring with identity and f(x) be a regular polynomial over A of degree n ≥ 1. Then there exists a ring extension K ⊇ A such that f(x) is a minimal polynomial over A of some element α 2 K. Proof. Let (1) be a regular polynomial over A. First, suppose that a0 = 1. If deg f(x) = n = 1, then we put α = −a1 and the statements are trivial. If n ≥ 2, then let K = A[y]/I be the quotient ring of the polynomial ring A[y] modulo the principal ideal I, generated by the polynomial f(y). Thus A¯ = fa + I j a 2 Ag is a subring of K and, because a0 = 1, it is clear that A and A are isomorphic rings. So A can be viewed as a subring of K. Obviously, f(x) is a minimal polynomial of the element α = y + I 2 K. If a0 6= 1, then let P = Q(A) be the classical ring of quotients of A. Now A ⊆ P , f(x) 2 P [x] and a0 is an invertible element of P . Therefore −1 g(x) = a0 f(x) is a monic polynomial over P and the statement for g(x) holds.
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